This threads shows the problems with labels, such as old school, new school, narration driven play etc. Any 2 people can agree on a definition, but once you add a third person to the discussion...
Anyway, yes you can run a megadungeon in 5E. Without tweaking rules. Many of us have done or are doing it.
Can you expand on that? I really don't think the PC-as-superhero, mechanically-anemic exploration pillar playstyle in WotC 5e is compatible with megadungeon play without some adjustments.
Labels... Cant say what you mean by those labels, but I would certainly never come close to describing a 5E character that way.
A couple things to keep in mind when approaching a mega dungeon campaign: sure, you can run it like I did when I was 8 and it's just ick in the door kill what's there. Or you can set it up as a living environment. Like DoMM tries to do or I did with my own version of Undermountain. Is it a dynamic environment, or is it static? Doesn't matter if you are running DoMM or LMoP, if you make the setting alive, then it's alive. Do things have reasons for being here and acting that way, or are they just fights to overcome?
Now, for all this rule tweaking debate. If you want supplies and long rests to be in limited supply, then make them so. There is always a reasonable in-game counter for anything that would say otherwise. (Wandering monsters, cursed zone, wild magic, etc). Worried about the magic hut? Make the inhabitants smart enough to learn (hence they have survived adventurers before). And remember, turn about is fair play. Don't limit yourself, and don't limit your players.